How are vectors and matrices multiplied in supervised machine learning?
In general, when people do not explicitly state it, a vector $v \in \mathbb{R}^n$ is usually considered a "column vector", that is, you can think of it as the matrix $v \in \mathbb{R}^{n \times 1}$ (that is, a matrix with $n$ rows and $1$ column). 
If it is not explicitly stated and you assume that the given vector is a column vector, but the dimensions do not match, then you should check that the dimensions match if you consider the given vector as a "row vector" (because it might be the case that the author is implicitly considering the vectors as row vectors).

Having said that, you can multiply a vector $v \in \mathbb{R}^n$ by a matrix $A \in \mathbb{R}^{n \times m}$ from the left, that is, you can do $v^T A$. Here, I considered $v$ has a column vector (that is, $v \in \mathbb{R}^{n \times 1}$), even though I have not explicitly stated it: you can and you often need to deduce this from the context! If I transpose $v$, I obtain $v^T \in \mathbb{R}^{1 \times n}$, and, thus, you can see that you can indeed perform the operation $v^T A = u \in \mathbb{R}^{1 \times m}$. Note that, at this point, $u$, the vector resulting from the operation $v^T A$, is actually considered a matrix, but you can still use it as a vector, if you need and that is permitted according to the mathematical rules of the operations you need to perform. Note also that I cannot multiply $v$ from the right of $A$, because there is no way of making the dimensions match. Have a look at this question, if you do not know how to multiply a vector by a matrix from the left.

Similarly, you can multiply $v \in \mathbb{R}^n$ by the matrix $B \in \mathbb{R}^{m \times n}$ only from the right. If $v$ is a column vector (that is, $v \in \mathbb{R}^{n \times 1}$), you need to do $B v \in \mathbb{R}^{m \times 1}$, but, if $v$ is a row vector (that is, $v \in \mathbb{R}^{1 \times n}$), you will first need to transpose it, so that you can perform the operation: $B v^T\in \mathbb{R}^{m \times 1}$.

Furthermore, note that, if you multiply a vector by a matrix from the left, that vector needs to be a "row vector", so, if you initially assume that the vector is a column vector (or that is explicitly stated), you will need to transpose it first, before the multiplication. However, if the vector is already a row vector, you won't have to transpose it. Similarly, if you multiply a vector by a matrix from the right, you will need a column vector.

To conclude, you can multiply a vector either from the left or right of a matrix, but you need to make sure that the dimensions match: if you multiply from the left, you will need to check that the dimensions of the vector match the number of the rows of the matrix; if you multiply the vector from the right of the matrix, you will need to check that the dimensions of the vector match the number of columns of the matrix.